2 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 2 Next, take the expectation: E[U(W)] = U(E[W]) + U (E[W])E [(W E[W])] U (E[W])E [ (W E[W]) 2] + i nfty 1 n! U(n) (E[W])E [(W E[W]) n ] n=3 With a mean-variance analysis we stop at the second order. There are two cases where this can be justified: If W is normally distributed, then the first two moments characterize all the moments. If the utility is quadratic, then U(W) (n) = 0 for n 3.

8 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 8 We want to characterize the mean-variance frontier (finding the portfolio with the lowest variance for a given expected return) The problem is This is a constrained optimization We write the Lagrangian min 2 w Σw subject to i w = 1 w 1 x w = µ

22 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 22 As long as a is not the minimum variance portfolio, it s possible to find a portfolio z that has a zero covariance with a: Cov(x a, x z ) = 1 a C x z = a 1 a + a x z A = 0 A C This is the expected return of a portfolio with zero covariance with any portfolio on the minimum variance frontier. We saw previously that V ar(x a ) = 1 a C + a A x a = a x z A + a A x a = a A ( x a x z ) using the result from previous slide

24 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 24 The last equality is the CAPM equation without a risk-free asset. It is telling us that the expected return on any portfolio p (i.e. x p ) is equal to the expected return on a portfolio uncorrelated with portfolio a (i.e. x z ) plus β pa times the excess return of a over z. Portfolio a is a portfolio on the minimum variance frontier. Portfolio z is a portfolio uncorrelated with portfolio a.

25 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management Introduction of a riskless asset Now assume there is one more asset. This asset is riskless and has a risk-free rate r f. The problem is now subject to 1 min w 2 w Σw w x + (1 w i)r f = µ

33 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 33 If we create a portfolio by combining the risk-free asset with a portfolio b on the frontier, we could get the following combination of µ and σ Expected return and standard deviation combination A/C b µ r f 0 0 1/C 1/2 σ

34 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 34 The portfolio x b would not be optimal. It is possible to get a higher µ for the same σ by switching from b to m (a portfolio that is tangent) Expected return and standard deviation combination m A/C µ r f 0 0 1/C 1/2 σ A combination of any other risky portfolio with the risk-free asset would give less µ for the same σ.

35 ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 35 It follows that everyone should choose a portfolio which falls on the r f m line. If you want higher pair (µ, σ), you put more weight on m. If you want lower pair (µ, σ), you put more weight on risk-free asset. The relative proportion of the risky assets should be the same regardless of where you are on the r f m line. We refer to m as the market portfolio. Your risk aversion will determine where on the r f m line you are * Higher risk aversion close to r f * Low risk aversion close to m or beyond m.

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